Brady was hurt in an accidental clash with Foxes defender Harry Maguire as he shaped to shoot on the edge of the box and left the field on a stretcher with his left leg in a brace.
Burnley manager Sean Dyche confirmed on Wednesday the Republic of Ireland winger has torn the patellar tendon in his knee and will need an operation on the injury.
"Robbie will have surgery to repair a patellar tendon tear and will be out for a substantial recovery period," Dyche said.
"He will be given all the time and care he needs on his recovery and supported throughout from all of us at Burnley Football Club."
Brady, who became the club's record signing when he joined from Norwich City for a reported £13 million in January 2016, had been in fine form for the high-flying Clarets, his thumping drive proving the difference in a 2-1 win at Bournemouth last week.
The 25-year-old's performances earned him a place on a seven-man shortlist for November's Player of the Month award, alongside the likes of Chelsea's Eden Hazard, Mohamed Salah of Liverpool and Manchester City duo Kevin De Bruyne and Raheem Sterling.
Dyche, who has led his side to seventh in the Premier League table, was also included on the shortlist for the manager prize, competing with Pep Guardiola, Jose Mourinho, Antonio Conte, Arsene Wenger and Jurgen Klopp for the award.